Abstract

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation (𝑝(𝑡)(𝑥Δ(𝑡))𝛾)Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0, on a time scale 𝕋, where 𝛾 is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.

1. Introduction

The theory of time scales was introduced by Hilger [1] in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Many authors have expounded on various aspects of this new theory, see [2–4]. A time scale 𝕋 is a nonempty closed subset of the real numbers, If the time scale equals the real numbers or integer numbers, it represents the classical theories of the differential and difference equations. Many other interesting time scales exist and give rise to many applications. The new theory of the so-called “dynamic equation” not only unify the theories of differential equations and difference equations, but also extends these classical cases to the so-called 𝑞-difference equations (when 𝕋=𝑞ℕ0∶={𝑞𝑡∶𝑡∈ℕ0 for 𝑞>1} or 𝕋=𝑞ℤ=𝑞ℤ∪{0}) which have important applications in quantum theory (see [5]). Also it can be applied on different types of time scales like 𝕋=ℎℤ,𝕋=ℕ20, and the space of the harmonic numbers 𝕋=𝕋𝑛. In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation (nonoscillation) of the solutions of different classes of dynamic equations on time scales, see [6–9]. In this paper, we deal with the oscillation behavior of all solutions of the second-order nonlinear delay dynamic equation𝑥𝑝(𝑡)Δ(𝑡)𝛾Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡0,(1.1)
subject to the hypotheses (H1)𝕋 is a time scale which is unbounded above, and 𝑡0∈𝕋 with 𝑡0>0. We define the time scale interval [𝑡0,∞)𝕋 by [𝑡0,∞)𝕋=[𝑡0⋂𝕋,∞).(H2)𝛾 is the quotient of odd positive integers.(H3)𝑝 and 𝑞 are positive rd-continuous functions on an arbitrary time scale 𝕋, and∞𝑡0Δ(𝑡)𝑝1/𝛾(𝑡)=∞(1.2)(H4)𝜏∶𝕋→𝕋 is a strictly increasing and differentiable function such that 𝜏(𝑡)≤𝑡, lim𝑡→∞𝜏(𝑡)=∞.(H5)𝑓∈𝐶(ℝ,ℝ) is a continuous function such that for some positive constant 𝐿, it satisfies 𝑓(𝑥)/𝑥𝛾≥𝐿 for all 𝑥≠0.

By a solution of (1.1), we mean that a nontrivial real valued function 𝑥 satisfies (1.1) for 𝑡∈𝕋. A solution 𝑥 of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. (1.1) is said to be oscillatory if all of its solutions are oscillatory. We concentrate our study to those solutions of (1.1) which are not identically vanishing eventually.

It is easy to see that (1.1) can be transformed into a half linear dynamic equation𝑥𝑝(𝑡)Δ(𝑡)𝛾Δ+𝑞(𝑡)𝑥𝛾(𝑡)=0,𝑡∈𝕋,𝑡≥𝑡0,(1.3)
where 𝑓(𝑥)=𝑥𝛾, 𝜏(𝑡)=𝑡. If 𝛾=1, then (1.1) is transformed into the equation𝑝(𝑡)𝑥Δ(𝑡)Δ+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡0.(1.4)
If 𝑝(𝑡)=1, then (1.4) has the form𝑥ΔΔ(𝑡)+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))=0,𝑡∈𝕋,𝑡≥𝑡0.(1.5)
If 𝑓(𝑥)=𝑥, then (1.5) becomes𝑥ΔΔ(𝑡)+𝑞(𝑡)𝑥(𝜏(𝑡))=0,𝑡∈𝕋,𝑡≥𝑡0.(1.6)
Recently, Zhang et al. [10] have considered the nonlinear delay (1.1) and established some sufficient conditions for oscillation of (1.1) when 𝛾≥1. Also Grace et al. [11] introduced some new sufficient conditions for oscillation of the half linear dynamic equation (1.3). In 2009, Sun et al. [12] extended and improved the results of [6, 13, 14] to (1.1) when 𝛾≥1, but their results can not be applied for 0<𝛾<1. In 2008, Hassan [15] considered the half linear dynamic equation (1.3) and established some sufficient conditions for oscillation of (1.3). In 2007, Erbe et al. [13] considered the nonlinear delay dynamic equation (1.4) and obtained some new oscillation criteria which improve the results of Şahiner [14]. In 2005, Agarwal et al. [6] studied the linear delay dynamic equation (1.6), also Şahiner [14] considered the nonlinear delay dynamic equation (1.5) and gave some sufficient conditions for oscillation of (1.6) and (1.5). In this work, we give some new oscillation criteria of (1.1) by using the generalized Riccati transformation and the inequality technique. Our results are general cases for some results of [12, 15].

This paper is organized as follows. In Section 2, we present some preliminaries on time scales. In Section 3, we give several lemmas. In Section 4, we establish some new sufficient conditions for oscillation of (1.1). Finally, in Section 5, we present some examples to illustrate our results.

2. Some Preliminaries on Time Scales

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. On any time scale 𝕋, we define the forward and backward jump operators by𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}.(2.1)
A point 𝑡∈𝕋, 𝑡>inf 𝕋 is said to be left dense if 𝜌(𝑡)=𝑡, right dense if 𝑡<sup𝕋 and 𝜎(𝑡)=𝑡, left scattered if 𝜌(𝑡)<𝑡, and right scattered if 𝜎(𝑡)>𝑡. The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡.

A function 𝑓∶𝕋→ℝ is called rd-continuous provided that it is continuous at right-dense points of 𝕋, and its left-sided limits exist (finite) at left-dense points of 𝕋. The set of rd-continuous functions is denoted by 𝐶rd(𝕋,ℝ). By 𝐶1rd(𝕋,ℝ), we mean the set of functions whose delta derivative belongs to 𝐶rd(𝕋,ℝ).

For a function 𝑓∶𝕋→ℝ (the range ℝ of 𝑓 may be actually replaced with any Banach space), the delta derivative 𝑓Δ is defined by 𝑓Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡)−𝑡,(2.2)
provided that 𝑓 is continuous at 𝑡, and 𝑡 is right scattered. If 𝑡 is not right scattered, then the derivative is defined by 𝑓Δ(𝑡)=lim𝑠→𝑡+𝑓(𝜎(𝑡))−𝑓(𝑡)𝑡−𝑠=lim𝑠→𝑡+𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠,(2.3)
provided that this limit exists.

A function 𝑓∶[𝑎,𝑏]→ℝ is said to be differentiable if its derivative exists. The derivative 𝑓Δ and the shift 𝑓𝜎 of a function 𝑓 are related by the equation 𝑓𝜎=𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓Δ(𝑡).(2.4)
The derivative rules of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔𝜎≠0) of two differentiable functions 𝑓 and 𝑔 are given by(𝑓⋅𝑔)Δ(𝑡)=𝑓Δ(𝑡)𝑔(𝑡)+𝑓𝜎(𝑡)𝑔Δ(𝑡)=𝑓(𝑡)𝑔Δ(𝑡)+𝑓Δ(𝑡)𝑔𝜎𝑓(𝑡),𝑔Δ𝑓(𝑡)=Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔Δ(𝑡)𝑔(𝑡)𝑔𝜎.(𝑡)(2.5)
An integration by parts formula reads𝑏𝑎𝑓(𝑡)𝑔Δ[](𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)𝑏𝑎−𝑏𝑎𝑓Δ(𝑡)𝑔𝜎(𝑡)Δ𝑡(2.6)
or 𝑏𝑎𝑓𝜎(𝑡)𝑔Δ[](𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)𝑏𝑎−𝑏𝑎𝑓Δ(𝑡)𝑔(𝑡)Δ𝑡(2.7)
and the infinite integral is defined by ∞𝑏𝑓(𝑠)Δ𝑠=lim𝑡→∞𝑡𝑏𝑓(𝑠)Δ𝑠.(2.8)
Note that in case 𝕋=ℝ, we have𝜎(𝑡)=𝜌(𝑡)=𝑡,𝜇(𝑡)=0,𝑓Δ(𝑡)=𝑓(𝑡),𝑏𝑎𝑓(𝑡)Δ𝑡=𝑏𝑎𝑓(𝑡)𝑑𝑡,(2.9)

and in case 𝕋=ℤ, we have𝜎(𝑡)=𝑡+1,𝜌(𝑡)=𝑡−1,𝜇(𝑡)=1,𝑓Δ(𝑡)=Δ𝑓(𝑡)=𝑓(𝑡+1)−𝑓(𝑡)if𝑎<𝑏,𝑏𝑎𝑓(𝑡)Δ𝑡=𝑏−1𝑡=𝑎𝑓(𝑡).(2.10)

Throughout this paper, we use𝑑+(𝑡)∶=max{0,𝑑(𝑡)},𝑑−𝛼(𝑡)∶=max{0,−𝑑(𝑡)},𝛽(𝑡)∶=𝛼(𝑡)0<𝛾≤1𝛾(𝑡)𝛾>1,(2.11)
where𝛼(𝑡)∶=𝑅(𝑡)𝑅(𝑡)+𝜇(𝑡),𝑅(𝑡)∶=𝑝1/𝛾(𝑡)𝑡𝑡0Δ𝑠𝑝1/𝛾(𝑠),for𝑡≥𝑡0.(2.12)

3. Several Lemmas

In this section, we present some lemmas that we need in the proofs of our results in Section 4.

Lemma 1 (Bohner and Peterson [3, Theorem 1.90]). If x(t) is delta differentiable and eventually positive or negative, then
((𝑥(𝑡))𝛾)Δ=𝛾10[]ℎ𝑥(𝜎(𝑡))+(1−ℎ)𝑥(𝑡)𝛾−1𝑥Δ(𝑡)𝑑ℎ.(3.1)

Lemma 2 (Hardy et al. [16, Theorem 41]). If 𝐴 and 𝐵 are nonnegative real numbers, then
𝜆𝐴𝐵𝜆−1−𝐴𝜆≤(𝜆−1)𝐵𝜆,𝜆>1,(3.2)
where the equality holds if and only if 𝐴=𝐵.

Lemma 3. If (H1)–(H3) and (1.2) hold and (1.1) has a positive solution 𝑥 on [𝑡0,∞)𝕋, then
𝑥𝑝(𝑡)Δ(𝑡)𝛾Δ<0,𝑥Δ(𝑡)>0,𝑥(𝑡)𝑥𝜎(𝑡𝑡)>𝛼(𝑡),for𝑡∈0,∞.(3.3)

Proof. The proof is similar to the proof of Lemma 2.1 in [15] and, hence, is omitted.

4. Main Results

Theorem 1. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1𝑟𝑑([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exists a positive Δ-differentiable function 𝛿(t) such that
limsup𝑡→∞𝑡𝑡0⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(s))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠=∞.(4.1)
Then every solution of (1.1) is oscillatory on [𝑡0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑡0,∞)𝕋. Then, without loss of generality, we assume that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡1,∞)𝕋,𝑡1∈[𝑡0,∞)𝕋, and there is 𝑇∈[𝑡0,∞)𝕋 such that 𝑥(𝑡) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Consider the generalized Riccati substitution
𝑥𝑤(𝑡)=𝛿(𝑡)𝑝(𝑡)Δ(𝑡)𝑥(𝜏(𝑡))𝛾.(4.2)
Using the delta derivative rules of the product and quotient of two functions, we have
𝑤Δ(𝑡)=𝛿Δ𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾(𝑥(𝜏(𝑡)))𝛾+𝛿𝜎𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾(𝑥(𝜏(𝑡)))𝛾Δ=𝛿Δ(𝑡)𝛿𝛿(𝑡)𝑤(𝑡)+𝜎𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾Δ(𝑥(𝜏(𝜎(𝑡))))𝛾−𝛿𝜎𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾((𝑥(𝜏(𝑡)))𝛾)Δ(𝑥(𝜏(𝑡)))𝛾(𝑥(𝜏(𝜎(𝑡))))𝛾,(4.3)
using the fact 𝑓(𝑥)/𝑥𝛾≥𝐿 and 𝑥(𝑡)/𝑥𝜎(𝑡)>𝛼(𝑡), we have
𝑤Δ(𝛿𝑡)≤Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝛿𝑡)𝑞(𝑡)−𝜎𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾((𝑥(𝜏(𝑡)))𝛾)Δ(𝑥(𝜏(𝑡)))𝛾(𝑥(𝜏(𝜎(𝑡))))𝛾.(4.4)
If 0<𝛾≤1, then using the chain rule and the fact that 𝑥(𝑡) is strictly increasing on [𝑇,∞)𝕋, we obtain
((𝑥(𝜏(𝑡)))𝛾)Δ=𝛾10𝑥(𝜏(𝑡))+ℎ𝜇(𝜏(𝑡))(𝑥(𝜏(𝑡)))Δ𝛾−1𝑑ℎ(𝑥(𝜏(𝑡)))Δ=𝛾10[(1−ℎ)𝑥(𝜏(𝑡))+ℎ𝑥𝜎](𝜏(𝑡))𝛾−1𝑑ℎ(𝑥(𝜏(𝑡)))Δ≥𝛾(𝑥𝜎(𝜏(𝑡)))𝛾−1(𝑥(𝜏(𝑡)))Δ𝜏Δ(𝑡)(4.5)
which implies
𝑤Δ𝛿(𝑡)≤Δ(𝑡)𝑤𝛿(𝑡)(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)−𝛾𝛿𝜎𝑥(𝑡)𝑝(𝑡)Δ(𝑡)𝛾(𝑥𝜎(𝜏(𝑡)))𝛾−1(𝑥(𝜏(𝑡)))Δ𝜏Δ(𝑡)(𝑥(𝜏(𝑡)))𝛾(𝑥(𝜏(𝜎(𝑡))))𝛾≤𝛿Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)−𝛾𝛿𝜎(𝑡)(𝑥(𝜏(𝑡)))Δ𝛼(𝜏(𝑡))𝜏Δ(𝑡)𝛿(𝑡)𝑥(𝜏(𝑡))𝑤(𝑡),(4.6)
since (𝑝(𝑡)(𝑥Δ(𝑡))𝛾)Δ<0, then by integrating from 𝑡 to 𝜏(𝑡), we get
(𝑥(𝜏(𝑡)))Δ>(𝑝(𝑡))1/𝛾((𝑝𝜏(𝑡)))1/𝛾𝑥Δ(𝑡)(4.7)𝑤Δ𝛿(𝑡)≤Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)q(t)−𝛾𝛿𝜎(𝑡)(𝑝(𝑡))1/𝛾𝑥Δ(𝑡)𝛼(𝜏(𝑡))𝜏Δ(𝑡)𝛿(𝑡)𝑥(𝜏(𝑡))(𝑝(𝜏(𝑡)))1/𝛾𝑤(𝑡),(4.8)
that is,
𝑤Δ≤𝛿(𝑡)Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)−𝛾𝛿𝜎(𝑡)𝛼(𝜏(𝑡))𝜏Δ(𝑡)𝛿(𝛾+1)/𝛾(𝑡)(𝑝(𝜏(𝑡)))1/𝛾𝑤(𝛾+1)/𝛾(𝑡).(4.9)If 𝛾>1, then using the chain rule and the fact that 𝑥(𝑡) is strictly increasing on [𝑇,∞)𝕋, we obtain
((𝑥(𝜏(𝑡)))𝛾)Δ≥(𝑥(𝜏(𝑡)))𝛾−1(𝑥(𝜏(𝑡)))Δ𝜏Δ(𝑡).(4.10)
From (4.4), (4.7), and (4.10), we have
𝑤Δ𝛿(𝑡)≤Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)−𝛾𝛿𝜎(𝑡)𝛼𝛾(𝜏(𝑡))𝜏Δ(𝑡)𝛿(𝛾+1)/𝛾(𝑡)(𝑝(𝜏(𝑡)))1/𝛾𝑤(𝛾+1)/𝛾(𝑡).(4.11)
By (4.9), (4.11), and the definition of 𝛽(𝑡), we have for 𝛾>0𝑤Δ𝛿(𝑡)≤Δ(𝑡)+𝛿(𝑡)𝑤(𝑡)−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)−𝛾𝛿𝜎(𝑡)𝛽(𝜏(𝑡))𝜏Δ(𝑡)𝛿𝜆(𝑡)𝑝𝜆−1𝑤(𝜏(𝑡))𝜆(𝑡),(4.12)
where 𝜆=(𝛾+1)/𝛾. Defining 𝐴≥0 and 𝐵≥0 by
𝐴𝜆=𝛾𝛿𝜎(𝑡)𝛽(𝜏(𝑡))𝜏Δ(𝑡)𝛿𝜆(𝑡)𝑝𝜆−1𝑤(𝜏(𝑡))𝜆(𝑡),𝐵𝜆−1=𝑝(𝜆−1)/𝜆𝛿(𝜏(𝑡))Δ(𝑡)+𝜆𝛾𝛿𝜎(𝑡)𝛽(𝜏(𝑡))𝜏Δ(𝑡)1/𝜆,(4.13)
then using Lemma 2, we get
𝛿Δ(𝑡)+𝛿(𝑡)𝑤(𝑡)−𝛾𝛿𝜎(𝑡)𝛽(𝜏(𝑡))𝜏Δ(𝑡)𝛿𝜆(𝑡)𝑝𝜆−1𝑤(𝜏(𝑡))𝜆𝛿(𝑡)≤𝑝(𝜏(𝑡))Δ(𝑡)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑡))𝛿𝜎(𝑡)𝜏Δ(𝑡)𝛾.(4.14)
From this last inequality and (4.12), we get
𝑤Δ(𝑡)≤−𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎𝛿(𝑡)𝑞(𝑡)+𝑝(𝜏(𝑡))Δ(𝑡)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑡))𝛿𝜎(𝑡)𝜏Δ(𝑡)𝛾.(4.15)
Integrating both sides from 𝑇 to 𝑡, we get
𝑡𝑇⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝛿𝜎𝛿(𝑠)𝑞(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠≤𝑤(𝑇)−𝑤(𝑡)≤𝑤(𝑇),(4.16)
which contradicts the assumption (4.1). This contradiction completes the proof.

Theorem 2. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1𝑟𝑑([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exist functions H,h∈C𝑟𝑑(𝔻,ℝ) (where 𝔻≡{(t,s)∶t≥s≥t0}) such that
𝐻(𝑡,𝑡)=0,𝑡≥𝑡0,𝐻(𝑡,𝑠)>0,𝑡>𝑠≥𝑡0,(4.17)
and 𝐻 has a nonpositive continuous Δ-partial derivative with respect to the second variable 𝐻Δ𝑠(𝑡,𝑠) which satisfies
𝐻Δ𝑠𝛿(𝜎(𝑡),𝑠)+𝐻(𝜎(𝑡),𝜎(𝑠))Δ(𝑡)𝛿(𝑡)=−ℎ(𝑡,𝑠)𝛿(𝑡)(𝐻(𝜎(𝑡),𝜎(𝑠)))𝛾/(𝛾+1),(4.18)limsup𝑡→∞1𝐻𝜎(𝑡),𝑡0𝑡𝜎(𝑡)0𝐾(𝑡,𝑠)Δ𝑠=∞,(4.19)
where 𝛿(𝑡) is positive Δ-differentiable function and
𝐾(𝑡,𝑠)=𝐻(𝜎(𝑡),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎ℎ(𝑠)−𝑝(𝜏(𝑠))−(𝑡,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾.(4.20)
Then every solution of (1.1) is oscillatory on [𝑡0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑡0,∞)𝕋. Then, without loss of generality, we assume that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡1,∞)𝕋,𝑡1∈[𝑡0,∞)𝕋, and there is 𝑇∈[𝑡0,∞)𝕋 such that 𝑥(𝑡) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Define 𝑤(𝑡) as in the proof of Theorem 1. Replacing (𝛿Δ(𝑡))+ with𝛿Δ (𝑡) in (4.12), we have
𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎(𝑡)𝑞(𝑡)≤−𝑤Δ𝛿(𝑡)+Δ(𝑡)𝛿(𝑡)𝑤(𝑡)−𝛾𝛿𝜎(𝑡)𝛽(𝜏(𝑡))𝜏Δ(𝑡)𝛿𝜆(𝑡)𝑝𝜆−1(𝑤𝜏(𝑡))𝜆(𝑡).(4.21)
Multiplying (4.21) by 𝐻(𝜎(𝑡),𝜎(𝑠)), and integrating with respect to 𝑠 from 𝑇 to 𝜎(𝑡), we get
𝑇𝜎(𝑡)𝐻(𝜎(𝑡),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))𝛿𝜎(𝑠)𝑞(𝑠)Δ𝑠≤−𝑇𝜎(𝑡)𝐻(𝜎(𝑡),𝜎(𝑠))𝑤Δ+(𝑠)Δ𝑠𝑇𝜎(𝑡)𝛿𝐻(𝜎(𝑡),𝜎(𝑠))Δ(𝑠)−𝛿(𝑠)𝑤(𝑠)Δ𝑠𝑇𝜎(𝑡)𝐻(𝜎(𝑡),𝜎(𝑠))𝛾𝛿𝜎(𝑠)𝛽(𝜏(𝑠))𝜏Δ(𝑠)𝛿𝜆(𝑠)𝑝𝜆−1𝑤(𝜏(𝑠))𝜆(𝑠)Δ𝑠.(4.22)
Integrating by parts and using (4.17) and (4.18), we obtain
𝑇𝜎(𝑡)𝐻(𝜎(𝑡),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))𝛿𝜎+(𝑠)𝑞(𝑠)Δ𝑠≤𝐻(𝜎(𝑡),𝑇)𝑤(𝑇)𝑇𝜎(𝑡)ℎ−(𝑡,𝑠)𝛿(𝑠)(𝐻(𝜎(𝑡),𝜎(𝑠)))1/𝜆𝑤(𝑠)−𝐻(𝜎(𝑡),𝜎(𝑠))𝛾𝛿𝜎(𝑠)𝛽(𝜏(𝑠))𝜏Δ(𝑠)𝛿𝜆(𝑠)𝑝𝜆−1𝑤(𝜏(𝑠))𝜆(𝑠)Δ𝑠.(4.23)
Defining 𝐴≥0 and 𝐵≥0 by
𝐴𝜆=𝐻(𝜎(𝑡),𝜎(𝑠))𝛾𝛿𝜎(𝑠)𝛽(𝜏(𝑠))𝜏Δ(𝑠)𝛿𝜆(𝑠)𝑝𝜆−1𝑤(𝜏(𝑠))𝜆(𝑠),𝐵𝜆−1=𝑝(𝜆−1)/𝜆(𝜏(𝑠))ℎ−(𝑡,𝑠)𝜆𝛾𝛿𝜎(𝑠)𝛽(𝜏(𝑠))𝜏Δ(𝑠)1/𝜆,(4.24)
then using Lemma 2, we get
ℎ−(𝑡,𝑠)𝛿(𝑠)(𝐻(𝜎(𝑡),𝜎(𝑠)))1/𝜆𝑤(𝑠)−𝐻(𝜎(𝑡),𝜎(𝑠))𝛾𝛿𝜎(𝑠)𝛽(𝜏(𝑠))𝜏Δ(𝑠)𝛿𝜆(𝑠)𝑝𝜆−1(𝑤𝜏(𝑠))𝜆≤ℎ(𝑠)𝑝(𝜏(𝑠))−(𝑡,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾,(4.25)
therefore,
𝑇𝜎(𝑡)𝐻(𝜎(𝑡),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))𝛿𝜎+(𝑠)𝑞(𝑠)Δ𝑠≤𝐻(𝜎(𝑡),𝑇)𝑤(𝑇)𝑇𝜎(𝑡)ℎ𝑝(𝜏(𝑠))−(𝑡,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾Δ𝑠.(4.26)
By the definition of 𝐾(𝑡,𝑠), we get
𝑇𝜎(𝑡)𝐾(𝑡,𝑠)Δ𝑠≤𝐻(𝜎(𝑡),𝑇)𝑤(𝑇),(4.27)
and this implies that
1𝐻(𝜎(𝑡),𝑇)𝑇𝜎(𝑡)𝐾(𝑡,𝑠)Δ𝑠≤𝑤(𝑇),(4.28)
which contradicts the assumption (4.19). This contradiction completes the proof.

Theorem 3. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1𝑟𝑑([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exists a positive Δ-differentiable function 𝛿(t) such that for m≥1limsup𝑡→∞1𝑡𝑚𝑡𝑡0(𝑡−𝑠)𝑚⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛿+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠=∞.(4.29)
Then every solution of (1.1) is oscillatory on [𝑡0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑡0,∞)𝕋. Then, without loss of generality, we assume that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡1,∞)𝕋,𝑡1∈[𝑡0,∞)𝕋, and there is 𝑇∈[𝑡0,∞)𝕋 such that 𝑥(𝑡) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Proceeding as in the proof of Theorem 1, we get (4.15) from which we have
𝐿𝛼𝛾(𝜏(𝑡))𝛿𝜎𝛿(𝑡)𝑞(𝑡)−𝑝(𝜏(𝑡))Δ(𝑡)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑡))𝛿𝜎(𝑡)𝜏Δ(𝑡)𝛾≤−𝑤Δ(𝑡),(4.30)
therefore,
𝑡𝑡1(𝑡−𝑠)𝑚⎛⎜⎜⎝𝐿𝛼𝛾(𝜏(𝑠))𝛿𝜎𝛿(𝑠)𝑞(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎞⎟⎟⎠Δ𝑠.≤−𝑡𝑡1(𝑡−𝑠)𝑚𝑤Δ(𝑡)Δ𝑠.(4.31)
The right hand side of the above inequality gives
𝑡𝑡1(𝑡−𝑠)𝑚𝑤Δ(𝑠)Δ𝑠=(𝑡−𝑠)𝑚𝑤(𝑠)𝑡𝑡1−𝑡𝑡1((𝑡−𝑠)𝑚)Δ𝑠𝑤(𝜎(𝑠))Δ𝑠.(4.32)
Since ((𝑡−𝑠)𝑚)Δ𝑠≤−𝑚(𝑡−𝜎(𝑠))𝑚−1≤0 for 𝑡≥𝜎(𝑠), 𝑚≥1, then we have
𝑡𝑡1(𝑡−𝑠)𝑚⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠≤𝑡−𝑡1𝑚𝑤𝑡1,(4.33)
then,
1𝑡𝑚𝑡𝑡1(𝑡−𝑠)𝑚⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠≤𝑡−𝑡1𝑡𝑚𝑤𝑡1(4.34)
which contradicts (4.29). This contradiction completes the proof.

Theorem 4. Assume that ∫∞𝑡0Δ𝑡/𝑝1/𝛾(𝑡)=∞ and
limsup𝑡→∞𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝑡𝑞(𝑠)Δ𝑠>1,ℎ𝑜𝑙𝑑.(4.35)
Then every solution of (1.1) is oscillatory on [𝑡0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑡0,∞)𝕋. Then, without loss of generality, we assume that 𝑥(𝑡)>0,𝑥(𝜏(𝑡))>0 for all 𝑡∈[𝑡1,∞)𝕋,𝑡1∈[𝑡0,∞)𝕋, and there is 𝑇∈[𝑡0,∞)𝕋 such that 𝑥(𝑡) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. From (1.1), we have
𝑥𝑝(𝑡)Δ(𝑡)𝛾Δ=−𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))≤−𝐿𝑞(𝑡)𝑥𝛾(𝜏(𝑡)).(4.36)
Integrating last equation from 𝜏(𝑡) to ∞, we obtain
∞𝜏(𝑡)𝐿𝑞(𝑠)𝑥𝛾𝑥(𝜏(𝑠))Δ𝑠<𝑝(𝜏(𝑡))Δ(𝜏(𝑡))𝛾−lim𝑠→∞𝑥𝑝(𝑠)Δ(𝑠)𝛾.(4.37)
Since 𝑝(𝑠)(𝑥Δ(𝑠))𝛾 decreasing and 𝑝(𝑠)(𝑥Δ(𝑠))𝛾>0, then we have
1𝑝(𝜏(𝑡))∞𝜏(𝑡)𝐿𝑞(𝑠)𝑥𝛾𝑥(𝜏(𝑠))Δ𝑠<Δ(𝜏(𝑡))𝛾.(4.38)
Since 𝑥(𝑡)>𝑅(𝑡)𝑥Δ(𝑡), then 𝑥(𝜏(𝑡))>𝑅(𝜏(𝑡))𝑥Δ(𝜏(𝑡)), and consequently
𝐿𝑝(𝜏(𝑡))∞𝜏(𝑡)𝑞(𝑠)𝑥𝛾(𝜏(𝑠))Δ𝑠<𝑥(𝜏(𝑡))𝑅(𝜏(𝑡))𝛾,𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝜏(𝑡)𝑞(𝑠)𝑥𝛾(𝜏(𝑠))Δ𝑠<𝑥𝛾(𝜏(𝑡)),(4.39)
but
𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝑡𝑞(𝑠)𝑥𝛾(𝜏(𝑠))Δ𝑠<𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝜏(𝑡)𝑞(𝑠)𝑥𝛾(𝜏(𝑠))Δ𝑠<𝑥𝛾(𝜏(𝑡)).(4.40)
Since 𝑥(𝑡) and 𝜏(𝑡) are strictly increasing, then we get that
𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝑡𝑞(𝑠)Δ𝑠<1,(4.41)
therefore,
𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝑡𝑞(𝑠)Δ𝑠≤1.(4.42)
This contradiction completes the proof.

5. Examples

In this section, we give some examples to illustrate our main results.

Example 1. Consider the second-order nonlinear delay dynamic equation
𝑡𝛾𝑥Δ(𝑡)𝛾Δ+𝜆𝑡𝛼𝛾𝑥(𝜏(𝑡))𝛾𝑡(𝜏(𝑡))=0for𝑡∈0,∞𝕋,𝑡0≥0,(5.1)
where 𝜆 is a positive constant,l and 𝛾 is the quotient of odd positive integers.
Here,
𝑝(𝑡)=𝑡𝛾𝜆,𝑞(𝑡)=𝑡𝛼𝛾(𝜏(𝑡)),𝑓(𝑥)=𝑥𝛾,𝐿=1.(5.2)
If 𝛿(𝑡)=1, then
∞𝑡0Δ𝑡(𝑝(𝑡))1/𝛾=∞𝑡0Δ𝑡𝑡=∞,limsup𝑡→∞𝑡𝑡0⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠=limsup𝑡→∞𝑡𝑡0𝜆𝑠Δ𝑠=∞.(5.3)
Therefore, by Theorem 1, every solution of (5.1) is oscillatory.

Example 2. Consider the second-order nonlinear delay dynamic equation
𝑥Δ(𝑡)𝛾Δ+𝜆𝜎𝛾−1(𝑡)𝑡𝛾𝜏𝛾(𝑥𝑡)𝛾𝑥(𝜏(𝑡))2𝛾𝑡(𝜏(𝑡))+1=0for𝑡∈0,∞𝕋,𝑡0≥0,(5.4)
where 𝜆 is a positive constant, and 0<𝛾≤1 is the quotient of odd positive integers, that is, 𝛼(𝑡)=𝛽(𝑡).
Here,
𝑝(𝑡)=1,𝑞(𝑡)=𝜆𝜎𝛾−1(𝑡)𝑡𝛾𝜏𝛾(𝑡),𝑓(𝑥)=𝑥𝛾𝑥2𝛾𝑡+1,𝐿=1,𝜏(𝑡)=2.(5.5)
It is clear that (1.2) holds.Since 𝑅(𝜏(𝑡))=𝑝1/𝛾∫(𝜏(𝑡))𝑡𝜏(𝑡)0Δ𝑠/𝑝1/𝛾(𝑠)=𝜏(𝑡)−𝑡0, then we can find 0<𝑏<1 such that
𝛼(𝜏(𝑡))=𝑅(𝜏(𝑡))=𝑅(𝜏(𝑡))+𝜇(𝜏(𝑡))𝜏(𝑡)−𝑡0𝜏(𝑡)−𝑡0=𝜏+𝜎(𝜏(𝑡))−𝜏(𝑡)(𝑡)−𝑡0𝜎(𝜏(𝑡))−𝑡0>𝑏𝜏(𝑡)𝜎(𝜏(𝑡)),for𝑡≥𝑡𝑏>𝑡0.(5.6)
If 𝛿(𝑡)=𝑡, then
limsup𝑡→∞𝑡𝑡0⎡⎢⎢⎣𝐿𝛼𝛾(𝜏(𝑠))𝑞(𝑠)𝛿𝜎𝛿(𝑠)−𝑝(𝜏(𝑠))Δ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))𝛿𝜎(𝑠)𝜏Δ(𝑠)𝛾⎤⎥⎥⎦Δ𝑠>limsup𝑡→∞𝑡𝑡0𝑏𝛾𝜏𝛾(𝑠)𝜆𝜎𝛾−1(𝑠)𝜎(𝑠)𝜎𝛾(𝜏(𝑠))𝜏𝛾(𝑠)𝑠𝛾−22𝛾𝜎𝛾(𝜏(𝑠))(𝛾+1)(𝛾+1)𝑏𝛾𝑠𝛾𝜎𝛾(𝑠)Δ𝑠>limsup𝑡→∞𝑡𝑡0𝑏𝛾𝜆𝜎𝛾(𝜏(𝑠))𝑠𝛾𝜎𝛾−2(𝜏(𝑠))2𝛾𝜎𝛾(𝑠)(𝛾+1)(𝛾+1)𝑏𝛾𝑠𝛾𝜎𝛾=𝑏(𝑠)Δ𝑠𝛾2𝜆−2𝛾(𝛾+1)(𝛾+1)𝑏𝛾limsup𝑡→∞𝑡𝑡01𝑠𝛾Δ𝑠=∞,(5.7)
if 𝜆>22𝛾/(𝑏2𝛾(𝛾+1)(𝛾+1)). Then by Theorem 1, every solution of (5.4) is oscillatory if 𝜆>22𝛾/(𝑏2𝛾(𝛾+1)(𝛾+1)).

Example 3. Consider the second-order nonlinear delay dynamic equation
𝑡𝛾−1𝑥Δ(𝑡)𝛾Δ+𝜆𝑥𝑡𝜎(𝑡)𝛾𝑡(𝜏(𝑡))=0for𝑡∈0,∞𝕋,𝑡0≥0,(5.8)
where 𝜆 is a positive constant and 𝛾≥1 is the quotient of odd positive integers.
Here,
𝑝(𝑡)=𝑡𝛾−1𝜆,𝑞(𝑡)=𝑡𝜎(𝑡),𝑓(𝑥)=𝑥𝛾,𝐿=1.(5.9)
It is clear that ∫∞𝑡0Δ𝑡/𝑝1/𝛾∫(𝑡)=∞𝑡0Δ𝑡/𝑡(𝛾−1)/𝛾=∞, for 𝛾≥1, (i.e., (1.2) holds) and 𝑅(𝜏(𝑡))≥𝜏(𝑡)−𝑡0≥𝑘𝜏(𝑡) for 0<𝑘<1, and 𝑡≥𝑡0≥1.
Then,
limsup𝑡→∞𝐿𝑅𝛾(𝜏(𝑡))𝑝(𝜏(𝑡))∞𝑡𝑞(𝑠)Δ𝑠≥limsup𝑡→∞𝑘𝛾𝜏𝛾(𝑡)𝜏𝛾−1(𝑡)∞𝑡𝜆𝑠𝜎(𝑠)Δ𝑠=𝜆limsup𝑡→∞𝑘𝛾𝜏(𝑡)∞𝑡−1𝑠ΔΔ𝑠=𝜆𝑘𝛾𝜏(𝑡)𝑡>1,(5.10)
if 𝜆>𝑡/𝑘𝛾𝜏(𝑡). Then by Theorem 4, every solution of (5.8) is oscillatory if 𝜆>𝑡/𝑘𝛾𝜏(𝑡).

Remarks 1. (1) The recent results due to Hassan [15], Grace et al. [11] and Agarwal et al. [7] cannot be applied to (5.1), (5.4), and (5.8) as they deal with ordinary equations without delay.(2) If 0<𝛾≤1, the results of Sun et al. [12] cannot be applied to (5.1) and (5.4).